The locus of points in a plane 1 unit from the point (0,0)

x² + y² = 1

Step-by-step explanation:

A circle is the locus of a point such that its distance from a fixed point (known as the origin) is always constant. The general equation of a circle is given as:

Where (h, k) is the center of the circle and r is the radius of the circle.

Given that the plane is 1 unit from the point (0,0). Therefore the radius of the circle is 1 unit and the center of the circle is (0, 0). Therefore the equation of the circle is:

the value of v.w= -12.

step-by-step explanation:

we are given three vector r,v and w as:

r = < 5, -5, -2> ; v = < 2, -8, -8> ; w = < -2, 6, -5>

we know that r,v and w could also be represented as:

r=5i-5j-2k

v=2i-8j-8k

and w= -2i+6j-5k

where i,j and k are the unit vectors of x,y and z-axis respectively.

also we know that it has the following property:

i.i=1 ; j.j=1; k.k=1

and i.=j.i=i.k=k.i=j.k=k.j=0

i.e. when we multiply two vectors or we find it's dot product we need to only multiply the components of x coordinates with each other; y with each other and z with each other.

hence, v.w= (2i-8j- -2i+6j-5k)

                  = 2×(-2)+(-8)×(6)+(-8)×(-5)

                  = -4-48+40

                  = -12

hence, v.w= -12